2 The Merton model: Equity as an option on a company’s assets
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value of the ﬁrm and the risk of default Chapter 39
and exactly the same partial differential equation for the equity of the ﬁrm S but with
S(A, T ) = max(A − D, 0).
Recognize this? The problem for S is exactly that for a call option, but now we have S
instead of the option value, the underlying variable are the assets A and the strike is D, the
debt. The solution for the equity value is precisely that for a call option.
39.2.1
Default Before Maturity
Creditors may be able to force liquidation of the company’s assets before maturity should the
value of its assets fall below a critical level. The critical level may be time-dependent:
A = K(t).
This would introduce a boundary condition on A = K(t) making this problem very similar
to that for a barrier option. On the (possibly moving) barrier the value of the debt is simply
whatever the assets are at that time:
V (K(t), t) = min(K(t), D).
39.2.2
Probability of Default
The probability of default before maturity P (A, t) is equivalent to the probability of the asset
value reaching the critical level K(t) before maturity:
∂P
∂ 2P
∂P
+ 12 σ 2 A2 2 + µA
=0
∂t
∂A
∂A
with
P (K(t), t) = 1 and P (A, T ) = 0.
39.2.3
Stochastic Interest Rates
We can make the model more realistic by introducing an interest rate model into the problem—after all, if there is no credit risk we would like to return to the simpler world of pricing
non-risky debt. I will be vague about the choice of interest rate model and just write
dr = u(r, t) dt + w(r, t) dX1 .
Still we will assume that
dA = µA dt + σ A dX2 .
There will be a correlation of ρ between the two random walks.
Now the value of the debt V , say, is a function of three variables; we have V (A, r, t). We
have seen in earlier chapters how to derive the equation for a bond value in the absence of
default risk. The result is a diffusion equation in r and t. The equation for V will be similar
but now there will be some A dependence and derivatives with respect to A.
641
642
Part Four credit risk
To ﬁnd the equation satisﬁed by V we construct a portfolio of our risky bond, and hedge it
with the stock and a riskless zero-coupon bond with price Z(r, t):
= V (A, r, t) −
S−
Z(r, t).
From this, we calculate d , choose and
to eliminate risk and set the return equal to the
risk-free rate. You know the drill, so cutting to the chase
∂V
∂ 2V
∂V
∂ 2V
∂ 2V
∂V
+ 12 w2 2 + ρσ wA
+ 12 σ 2 A2 2 + (u − λw)
+ rA
− rV = 0
∂t
∂r
∂r∂A
∂A
∂r
∂A
where λ is again the price of interest rate risk.
The ﬁnal condition for this equation is
V (A, T ) = min(D, A)
representing the payment of the debt at maturity.
The main criticism of this model is that it is very difﬁcult to measure variables and parameters.
Nevertheless, it can be useful as a phenomenological model, perhaps for estimating the relative
value of different types of debt issued by the same business, or businesses with the same credit
rating.
39.3 MODELING WITH MEASURABLE
PARAMETERS AND VARIABLES
In this section I describe a model that takes easily measurable
inputs. We will concentrate on valuing the debt when interest
rates are deterministic; the model could easily be extended to
stochastic interest rates at the cost of additional complexity and
computing time.
We will value the debt of a company having a very simple
operating procedure: They sell their product, they pay their costs and they put any proﬁt into the
bank. The key quantity we will model is the earnings of the company. Think of these earnings
as being the gross income from the sale of the product. The net earnings or proﬁt will be the
gross earnings less the costs. Assume that the gross annualized earnings E of the company are
random:
dE = µE dt + σ E dX.
(Of course, we need not choose a lognormal model, but it is the traditional starting point.)
We assume that the company has ﬁxed costs of E ∗ per annum and ﬂoating costs of kE. The
proﬁt of E − E ∗ − kE = (1 − k)E − E ∗ is put into a bank earning a ﬁxed rate of interest r.
If we denote by C the cash in this bank account then this is given by
t
C=
(1 − k)E(τ ) − E ∗ er(t−τ ) dτ .
0
This expression represents the accumulation of income together with any bank interest. Differentiating this gives the stochastic differential equation satisﬁed by C:
dC = (1 − k)E − E ∗ + rC dt.
value of the ﬁrm and the risk of default Chapter 39
I have chosen to model the earnings of the business rather than the ﬁrm’s value since the
former are far easier to measure, requiring only an examination of the ﬁrm’s accounts perhaps.
We shall see how the value of the ﬁrm is then an output of the model.
The well-being of the company is determined by its earnings and bank account balance at
any time, i.e. by E and C. The owners of the company hope that (1 − k)E > E ∗ , but even if
it is not (at the start of trading, say), then perhaps the growth in earnings will eventually take
the company into proﬁt.
Suppose that the company owes D which must be repaid at time T . We make the simple
assumption that if the company has D in the bank at time T then it will repay the debt, if it has
less than D in the bank it will repay everything that it has, and if there is a negative amount
in the bank then they repay nothing. This gives a repayment of
max(min(C, D), 0).
(39.1)
Again, this would be more complicated if we were to incorporate partial repayment or reﬁnancing.
The value of the debt will be a function of E, C and t. Introduce the quantity V (E, C, t)
as the present value of the expected amount in (39.1). This function satisﬁes the differential
equation
∂V
∂V
∂V
∂ 2V
+ 12 σ 2 E 2 2 + µE
+ (1 − k)E − E ∗ + rC
− rV = 0,
∂t
∂E
∂E
∂C
with ﬁnal condition
V (E, C, T ) = max(min(C, D), 0).
In Figure 39.1 we see a plot of the value of the debt according to this model when there is
an amount $100,000 to be repaid in two years, with a risk-free interest rate of 5%. The ﬁrm
has ﬁxed costs of $30,000 p.a. and variable costs of 7%. The drift of the earnings is 10% and
the volatility 25%. Observe that for both large C and E the value approaches the value of a
risk-free zero-coupon bond. In Figure 39.2 is the equivalent yield, deﬁned as
− 12 log
V
.
100,000
The 12 being in front since it is a two-year loan. Subtract the risk-free 5% from this and you
get the credit spread.
As it stands this problem can be usefully used to value risky debt: The value of the debt is
simply V (E, C, t) with today’s values for E, C and t. However, it can be modiﬁed quite simply
to accommodate more sophisticated operating procedures for the company. One possibility is to
say that the company closes down immediately that it goes into the red. This could be modeled
by the boundary condition
V (E, 0, t) = 0.
643
Part Four credit risk
100000
90000
70000
60000
50000
60000
54000
48000
42000
36000
30000
Earnings
24000
18000
12000
40000
30000
20000
Present value of debt
80000
10000
0
−80000
−40000
0
40000
80000
120000
160000
200000
240000
280000
320000
360000
400000
−120000
−320000
−280000
−240000
−200000
−160000
−10000
6000
0
−400000
−360000
Cash
Figure 39.1 Value of debt issued by a limited liability company as a function of annual earnings
and retained cash.
25%
20%
15%
10%
5%
Figure 39.2 Effective yield.
Cash
54000
48000
42000
36000
30000
18000
120000
12000
200000
6000
Earnings
24000
280000
60000
0%
360000
0
644
value of the ﬁrm and the risk of default Chapter 39
39.4
CALCULATING THE VALUE OF THE FIRM
By changing ﬁnal and boundary conditions it is a very simple matter to use the above model
to value the ﬁrm, and to examine the effects of various business strategies on that value.
For example, suppose that we take the value of the business to be the present value of the
expected cash in the bank at some time T0 in the future. Such a ﬁnite time horizon is a
common assumption when we are estimating the present value of a potentially inﬁnite sum of
cashﬂows which are (one hopes) growing faster than the interest rate. In this case, the ﬁrm
value V (E, C, t) satisﬁes
∂V
∂V
∂V
∂ 2V
+ 12 σ 2 E 2 2 + µE
+ (1 − k)E − E ∗ + rC
− rV = 0.
∂t
∂E
∂E
∂C
As an example of the ﬂexibility of this approach, consider the different ﬁnal conditions applying
to the two different problems valuing a limited liability company and valuing an unlimited
liability partnership.
Limited liability
If the business has no liability when it has a negative amount in the bank at time T0 , then
V (E, C, T0 ) = max(C, 0).
Partnership
If the owners of the business are liable for the debts of the business then
V (E, C, T0 ) = C.
In the former case, if the business expires in the red then the company directors declare
bankruptcy and walk away from the debt (assuming that they have not acted negligently).
Not only can we use this model to examine the legal standing of the company, but also to
study the effects of various operating procedures on its value. Here is an example.
Optimal close-down
If the model gives a value of $3,000,000 to your company but you currently have $5,000,000 in
the bank then the model is trying to tell you something: Bad times are just around the corner. In
such a situation it is not wise to keep trading; you will be better off closing down the business.
The decision to close down the business can be optimized by a constraint of the form
V (E, C, t) ≥ C
with continuity of the ﬁrst derivatives. This is just like an American option problem and the
justiﬁcation is similar. An example of the company valuation problem with this constraint is
shown in Figure 39.3.
645
Part Four credit risk
700000
600000
500000
400000
300000
200000
100000
−4.E+04
2.E+04
8.E+04
1.E+05
2.E+05
3.E+05
3.E+05
4.E+05
−1.E+05
−2.E+05
−3.E+05
−3.E+05
Earnings
−2.E+05
0
60000
57000
54000
51000
48000
45000
42000
39000
36000
33000
30000
27000
24000
21000
18000
15000
12000
9000
6000
3000
−4.E+06
646
Cash
Figure 39.3 Company valuation with optimal close-down. Parameter values are the same as in
the previous ﬁgure.
39.5 SUMMARY
I’ve introduced ﬁrm valuation via a credit risk problem. There are many other reasons why one
might want to know the value of a ﬁrm and these are discussed in depth in Chapter 73. But
next, in Chapter 40 we examine another approach to modeling credit risk, one that supposedly
requires no detailed knowledge of the ﬁrm issuing the debt.
FURTHER READING
• See Black & Scholes (1973), Merton (1974), Black & Cox (1976), Geske (1977) and Chance
(1990) for a treatment of the debt of a ﬁrm as an option on the assets of that ﬁrm. See
Longstaff & Schwartz (1994) for more recent work in this area.
• See Apabhai et al. (1998) for more details of the company and debt valuation model,
especially for ﬁnal and boundary conditions for various business strategies. Epstein et al.
value of the ﬁrm and the risk of default Chapter 39
(1997a, b) describe the ﬁrm valuation models in detail, including the effects of advertising
and market research.
• The classic reference, and a very good read, for ﬁrm-valuation modeling is Dixit & Pindyck
(1994). For a non-technical POV see Copeland, Koller & Murrin (1990).
• See Kim (1995) for the application of the company valuation model to the question of
company mergers and some suggestions for how it can be applied to problems in company
relocation and tax status.
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CHAPTER 40
credit risk
In this Chapter. . .
•
•
•
•
•
models for instantaneous and exogenous risk of default
stochastic risk of default and implied risk of default
credit ratings
how to model change of rating
how to model risk of default in convertible bonds
40.1
INTRODUCTION
In the previous chapter I described some ways of looking at default via models for the creditworthiness of the ﬁrm issuing the debt. This is a nice approach if you have access to all the
data. A more recent approach is to model default as a completely exogenous event i.e. a bit
like the tossing of a coin or the appearance of zero on a roulette wheel, and having nothing to
do with how well the company or country is doing. Typically, one then infers from risky bond
prices the probability of default as perceived by the market. I’m not wild about this idea but it
is very popular.
Later in this chapter I describe the rating service provided by Standard & Poor’s and Moody’s,
for example. These ratings provide a published estimate of the relative creditworthiness of ﬁrms.
40.2
RISKY BONDS
If you are a company wanting to expand, but without the necessary capital, you could borrow
the capital, intending to pay it back with interest at some time in the future. There is a chance,
however, that before you pay off the debt the company may have got into ﬁnancial difﬁculties
or even gone bankrupt. If this happens, the debt may never be paid off. Because of this risk of
default, the lender will require a signiﬁcantly higher interest rate than if he were lending to a
very reliable borrower such as the US government.
The real situation is, of course, more complicated than this. In practice it is not just a case
of all or nothing. This brings us to the ideas of the seniority of debt and the partial payment
of debt.
Firms typically issue bonds with a legal ranking, determining which of the bonds take priority
in the event of bankruptcy or inability to repay. The higher the priority, the more likely the debt
Part Four credit risk
12
10
8
Yield
650
6
US curve
Ba3
B1
Caa1
NR
4
2
0
0
2
4
6
8
10
12
Duration
Figure 40.1 Yield versus duration for some risky bonds.
is to be repaid, the higher the bond value and the lower its yield. In the event of bankruptcy
there is typically a long, drawn out battle over which creditors get paid. It is usual, even after
many years, for creditors to get some of their money back. Then the question is how much and
how soon? It is also possible for the repayment to be another risky bond; this would correspond
to a reﬁnancing of the debt. For example, the debt could not be paid off at the planned time so
instead a new bond gets issued entitling the holder to an amount further in the future.
In Figure 40.1 is shown the yield versus duration, calculated by the methods of Chapter 13,
for some risky bonds. In this ﬁgure the bonds have been ranked according to their estimated
riskiness. We will discuss this later, for the moment you just need to know that Ba3 is considered
to be less risky than Caa1 and this is reﬂected in its smaller yield spread over the risk-free curve.
The problem that we examine in this chapter is the modeling of the risk of default and thus
the fair value of risky bonds. Conversely, if we know the value of a bond, does this tell us
anything about the likelihood of default?
40.3 MODELING THE RISK OF DEFAULT
The models that I have described or will describe here fall into two categories, those for which
the likelihood of default depends on the behavior of the issuing ﬁrm and those for which the
likelihood of default is exogenous. The former is appealing because it is clearly closer to reality.
The downside is that these models are usually more complicated to solve, with parameters that
are difﬁcult to measure.
The instantaneous risk of default models are simpler to use and are therefore the most popular
type of credit risk models. In its simplest form the time at which default occurs is completely
exogenous. For example, we could roll a die once a month, and when a 1 is thrown the company
defaults. This illustrates the exogeneity of the default and also its randomness; a Poisson process
is a typical choice for the time of default. We will see that when the intensity of the Poisson
process is constant (as in the die example), the pricing of risky bonds amounts to the addition
of a time-independent spread to the bond yield. We will also see models for which the intensity
is itself a random variable.
credit risk Chapter 40
A reﬁnement of the modeling that we also consider is the regrading of bonds. There are agents,
such as Standard & Poor’s and Moody’s, who classify bonds according to their estimate of
their risk of default. A bond may start life with a high rating, for example, but may ﬁnd itself
regraded due to the performance of the issuing ﬁrm. Such a regrading will have a major effect
on the perceived risk of default and on the price of the bond. I will describe a simple model
for the rerating of risky bonds.
40.4
THE POISSON PROCESS
AND THE INSTANTANEOUS
RISK OF DEFAULT
A popular approach to the modeling of credit risk is via the
instantaneous risk of default, p. If at time t the company has
not defaulted and the instantaneous risk of default is p then the
probability of default between times t and t + dt is p dt. This
is an example of a Poisson process, as described in detail in Chapter 57; nothing
happens for a while, then there is a sudden change of state. This is a continuous-time
version of our earlier model of throwing a die.
The simplest example to start with is to take p constant. In this case we can easily
determine the risk of default before time T . We do this as follows.
Let P (t; T ) be the probability that the company does not default before time T
given that it has not defaulted at time t. The probability of default between later times
t and t + dt is the product of p dt and the probability that the company has not
defaulted up until time t . Thus,
P (t + dt , T ) − P (t , T ) = p dt P (t , T ).
Expanding this for a small time step results in the ordinary differential equation
representing the rate of change of the required probability:
∂P
= pP (t ; T ).
∂t
If the company starts out not in default then P (T ; T ) = 1. The solution of this problem is
e−p(T −t) .
The value of a zero-coupon bond paying $1 at time T could therefore be modeled by taking
the present value of the expected cashﬂow. This results in a value of
e−p(T −t) Z,
(40.1)
where Z is the value of a riskless zero-coupon bond of the same maturity as the risky bond.
Note that this does not put any value on the risk taken. The yield to maturity on this bond is
now given by
log(e−p(T −t) Z)
log Z
−
=−
+ p.
T −t
T −t
Thus the effect of the risk of default on the yield is to add a spread of p. In this simple model,
the spread will be constant across all maturities.
651